Problem: $f(x,y) = \sin(x^2)$ What is $\dfrac{\partial f}{\partial x}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A $0$ (Choice B) B $\cos(x^2)$ (Choice C) C $2xy\cos(x^2)$ (Choice D) D $2x\cos(x^2)$
Solution: We want to find $\dfrac{\partial f}{\partial x}$, which is the partial derivative of $f$ with respect to $x$. When we take a partial derivative with respect to $x$, we treat $y$ as if it were a constant. Here, $f(x, y)$ only has one term. $\begin{aligned} &\dfrac{\partial}{\partial x} \left[ \sin^2(x) \right] = 2x\cos(x^2) \end{aligned}$ In conclusion: $\dfrac{\partial f}{\partial x} = 2x\cos(x^2)$